Deriving Lehmer and H\"older means as maximum weighted likelihood estimates for the multivariate exponential family

TL;DR

This paper extends the connection between Lehmer and H"older means and maximum likelihood estimators from univariate to multivariate exponential families, offering a new probabilistic interpretation.

Contribution

It generalizes the relationship between these means and maximum likelihood estimates to multivariate cases, broadening their theoretical understanding and potential applications.

Findings

Established the link for multivariate exponential families

Provided a probabilistic interpretation of Lehmer and H"older means

Extended previous univariate results to multivariate setting

Abstract

The links between the mean families of Lehmer and H\"older and the weighted maximum likelihood estimator have recently been established in the case of a regular univariate exponential family. In this article, we will extend the outcomes obtained to the multivariate case. This extension provides a probabilistic interpretation of these families of means and could therefore broaden their uses in various applications.